Is there a good way to make a "perfect" dome?
You can instead just use the boolean modifier in order to cut away the quad sphere. It should still keep all-quad geometry but the faces near the cut could be squished.
Important notes:

Using subdivision surface on a cube will not give you a perfect sphere. The corners will always be slightly lopsided.
Instead: Enable the &quot;Extra Objects- Mesh&quot; addon inside Edit-&gt;Preferences-&gt;Addons. This comes with Blender. Then, when you add a new object you can choose &quot;Round Cube&quot; inside the mesh section. Choose that, and switch the operator preset to &quot;quadsphere&quot; to make a truly round quadsphere.